# Boolean Pythagorean triples problem

- kind: problem
- status: solved (evidence tier V3)
- year: 1980
- authors: Ronald Graham

**Statement.** Can the positive integers be colored red and blue so that no Pythagorean triple is monochromatic?

No: the integers 1 to 7,825 cannot be colored that way, while 1 to 7,824 can. A SAT solver proved it and produced a 200-terabyte certificate that independent checkers verified. Graham paid the $100 prize he had offered. It is the pure form of a machine-verified prize.

*Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.*

Posed 1980s, Ronald Graham. Closed 2016, Marijn Heule, Oliver Kullmann and Victor Marek.

## What would settle it

DRAT proof certificate, independently checked

## Prize

$100 from Ronald Graham, paid

## Sources

- [Wikipedia: Boolean Pythagorean triples problem](https://en.wikipedia.org/wiki/Boolean_Pythagorean_triples_problem)

[Combinatorics](/wiki/Space:combinatorics.md) · [Mathematics](/wiki/Continent:mathematics.md)

---
Revision 39, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:boolean-pythagorean-triples?action=history. Edit: POST /api/wiki/page/Problem:boolean-pythagorean-triples {source, summary, base_rev: 39}.